Over the years, a number of approaches have been suggested to treat the small cell problem in the context of cut cell meshes for single phase problems, see, e.g.,
Refs. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. Note that several of the suggested approaches rely on unifying the solution values of the small cell and their bigger neighbors in a suitable way. This is not a feasible option for two phase flow problems. We therefore follow the mixed explicit implicit approach here, where standard explicit time stepping is used away from the small cell and implicit time stepping is used in the neighborhood of the small cell for stability. A similar approach of combining explicit and implicit time stepping has been used very recently by Fu et al.[10]} in the context of locally moving interfaces for the advection problem for higher-order discontinuous Galerkin methods.
